Characterization of regularity for a connected Abelian action

Let V be a finite dimensional real vector space, let g be the real span of a finite set of commuting endomorphisms of V, and G = exp g. We study the orbit structure in elements of a finite partition of V into explicit G-invariant connected sets. In particular, we prove that either there is an open conull G-invariant subset Omega of V in which every G-orbit is regular, or there is a G-invariant, conull, G(delta) subset of V in which every orbit is not regular. We present an explicit computable necessary and sufficient condition for almost everywhere regularity. Finally in the case of regularity we construct an explicit topological cross-section for the orbits in Omega.